Minkowski Space-Time: a Glorious Non-Entity

DRAFT: written for Petkov (ed.), The Ontology of Spacetime (in preparation)

Minkowski space-time: a glorious non-entity

Harvey R Brown∗ and Oliver Pooley† 16 March, 2004

Abstract It is argued that Minkowski space-time cannot serve as the deep structure within a “constructive” version of the special theory of relativity, contrary to widespread opinion in the philosophical community.

This paper is dedicated to the memory of Jeeva Anandan.

Contents

1 Einstein and the space-time explanation of inertia 1

2 The nature of absolute space-time 3

3 The principle vs. constructive theory distinction 4

4 The explanation of length contraction 8

5 Minkowski space-time: the cart or the horse? 12

1 Einstein and the space-time explanation of inertia

It was a source of satisfaction for Einstein that in developing the general theory of relativity (GR) he was able to eradicate what he saw as an embarrassing defect of his earlier special theory (SR): violation of the action-reaction principle. Leibniz held that a defining attribute of substances was their both acting and being acted upon. It would appear that Einstein shared this view. He wrote in 1924 that each physical object “influences and in general is influenced in turn by others.”1 It is “contrary to the mode of scientific thinking”, he wrote earlier in 1922, “to conceive of a thing. . . which acts itself, but which cannot be acted upon.”2 But according to Einstein the space-time continuum, in both Newtonian mechanics and special relativity, is such a thing. In these theories

∗Faculty of Philosophy, University of Oxford, 10 Merton Street, Oxford OX1 4JJ, U.K.; [email protected] †Oriel College, Oxford OX1 4EW, U.K.; [email protected] 1Einstein (1924, 15). 2Einstein (1922, 55–6). For a recent discussion of the action-reaction principle in modern physics, see Anandan and Brown (1995) and Brown (1996).

1 space-time upholds only half of the bargain: it acts on material bodies and/or fields, but is in no way influenced by them. It is important to ask what kind of action Einstein thought is involved here. Although he did not describe them in these terms, it seems that he had in mind the roles of the four-dimensional absolute affine connection in each case, as well as that of the conformal structure in SR. The connection determines which paths are geodesics, or straight, and hence determines the possible trajectories of force- free bodies. The null cones in SR in turn constrain the possible propagation of light.

The inertia-producing property of this ether [Newtonian space-time], in accordance with classical mechanics, is precisely not to be influenced, either by the configuration of matter, or by anything else. For this reason, one may call it “absolute”. That something real has to be conceived as the cause for the preference of an inertial system over a noninertial system is a fact that physicists have only come to understand in recent years ... Also, following the special theory of relativity, the ether was absolute, because its influence on inertia and light propagation was thought to be independent of physical influences of any kind ... The ether of the general theory of relativity differs from that of classical mechanics or the special theory of relativity respectively, insofar as it it is not “absolute”, but is determined in its locally variable properties by ponderable matter. (Einstein, 1924)

The success in salvaging the action-reaction principle was not confined in GR to the fact that the space-time metric field (which of course determines both the connection, by the principle of metric compatibility, and the conformal structure) is dynamical, being a solution to Einstein’s field equations, which couple matter degrees of freedom to the metric. In the early twenties, when he wrote the above comment, Einstein had still not discovered an important aspect of his theory of gravitation—the fact that the field equations themselves underpin the geodesic principle. This principle states that the world-lines of force-free test particles are constrained to lie on geodesics of the connection. It is important for our purposes to dwell briefly on the significance of this fact. We have seen that for Einstein the inertial property of matter3 requires explanation in terms of the action of a real entity on the particles. It is the space-time connection that plays this role: the affine geodesics form ruts or grooves in space-time that guide the free particles along their way. The intuition was well expressed by Nerlich in 1976: . . . without the affine structure there is nothing to determine how the [free] particle trajectory should lie. It has no antennae to tell it where other objects are, even if there were other objects. . . It is because space-time has a certain shape that world lines lie as they do. (Nerlich, 1976, 264, original emphasis) In GR, on the other hand, this view is at best redundant, at worst problem- atic. For it follows from the form of Einstein’s field equations that the covariant

3To be precise, the fact that particles with non-zero mass satisfy Newton’s ﬁrst law of motion, not that they possess such inertial mass

2 divergence of the stress-energy tensor ﬁeld Tµν —that object which incorporates the “matter” degrees of freedom—vanishes.

µ T ν;µ = 0 (1) This result is about as close as anything is in GR to the statement of a con- servation principle, and it came to be recognised as the basis of a proof, or proofs, that the world-lines of a suitably modelled force-free test particles are geodesics.4 The fact that these proofs vary considerably in detail need not de- tain us. The first salient point is that the geodesic principle for free particles is no longer a postulate but a theorem. GR is the first in the long line of dynamical theories, starting with the Aristotelian system and based on that profound distinction between natural and forced motions of bodies, that explains inertial motion. The second point is that the derivations of the geodesic principle in GR also demonstrate its limited validity. In particular, it is not enough that the test particle be force-free. It has long been recognised that spinning bodies for which tidal gravitational forces act on its elementary pieces deviate from geodesic behaviour.5 What this fact should clarify, if indeed clarification is needed, is that it is not simply in the nature of force-free bodies to move in a fashion consistent with the geodesic principle. It is not an essential property of localised bodies that they run along the ruts of space-time determined by the affine connection, when no other dynamical influences are at play. In Newtonian mechanics and SR, the conspiracy of inertia is a postulate, and its putative explanation by way of the affine connection is a postulate added to a postulate. And it is here that Einstein and Nerlich part company with Leibniz, and even Newton. For both Leibniz and Newton, absolute space-time structure is not the sort of thing that acts at all. If this is correct, and we believe it is, then neither Newtonian mechanics nor SR represent, pace Einstein, a violation of the action-reaction principle, because the space-time structures in both cases are neither acting nor being acted upon. Indeed we go further and agree with Leibniz that they are not real entities in their own right at all.

2 The nature of absolute space-time

It is well known that Leibniz rejected the reality of absolute Newtonian space and time principally on the grounds that their existence would clash with his principles of Sufficient Reason and the Identity of Indiscernibles. Nonentities do not act, so for Leibniz space and time can play no role in explaining the mystery of inertia. 4See, for example, Misner et al. (1973, §20.6, 471–80). 5See Misner et al. (1973, 480; ex. 40.8, 1120–1; and §40.9, 1126–31). These authors refer briefly on p. 480 to the complications that quantum physics is likely to introduce to the question of geodesic behaviour. We note that the familiar picture of light tracing out the null cones of space-time is also probably only approximately valid (though the approximation is usually extremely good) as a result of quantum physics. Since 1980, studies have been made of the propagation of photons in QED in curved space-times, in the esoteric regime where the scale of the space-time curvature is comparable to the Compton wavelength of the electron. Here, vacuum polarisation causes the vacuum to act both as a dispersive and birefringent optical medium. In particular the propagation of photons as determined by geometric optics is controlled by an effective metric that differs from the space-time metric gµν . For a recent review paper, see Shore (2003).

3 Newton seems to have agreed with this conclusion, but for radically different reasons, as expressed in his pre-Principia manuscript De Gravitatione (Newton, 1962). For Newton, the existence of absolute space and time has to do with providing a structure, necessarily distinct from ponderable bodies and their relations, with respect to which it is possible systematically to define the basic kinematical properties of the motion of such bodies. For Newton, space and time are not substances in the sense that they can act, but are real things nonetheless.6 It is now known, however, that the job can be done without pos- tulating any background space-time scaffolding, and that at least a significant subset—perhaps the significant subset—of solutions to any Newtonian theory can be recovered in the process.7 Recall Nerlich’s remark above to the effect that force-free particles have no antennae, that they are unaware of the existence of other particles. That is the prima facie mystery of inertia in pre-GR theories: how do all the free particles in the world know how to behave in a mutually coordinated way such that their motion appears extremely simple from the point of view of a family of privileged frames? But to appeal to the action of a background space-time connection in which the particles are immersed—to what Weyl called the “guiding field”—is arguably to enhance the mystery, not to remove it. For the particles do not have space-time feelers either. In what sense is the postulation of the 4-connection doing more explanatory work than Moliere’s famous dormative virtue in opium? (We return to this question below.) It is of course non-trivial that inertia can be given a geometrical description, and this is associated with the fact that the behaviour of force-free bodies does not depend on their constitution: it is universal. But again what is at issue is the arrow of explanation. In our view it is simply more economical to consider the 4-connection as a codification of certain key aspects of the behaviour of particles and fields.8

3 The principle vs. constructive theory distinction

In recent years there has been increasing discussion of the role that thermodynamics played as a methodological template in Einstein’s development of special relativity, and of his characterization of SR as a “principle” theory, as opposed to a “constructive” theory like the kinetic theory of gases.9 The distinction is not a categorical one, nor must a principle theory be bereft of any constructive elements. What we have eﬀectively argued in section 1 is

6It is worth stressing that its lack of causal influence is Newton’s sole reason for refraining from calling space a substance. It is therefore at least misleading to deny that Newton was a substantivalist. 7The discovery was made by Julian Barbour and Bruno Bertotti (Barbour and Bertotti, 1982; Barbour, 1994). For discussion see Belot (2000) and Pooley and Brown (2002). 8One faces a similar choice in parity-violating theories: do orientation fields play an explanatory role in such theories, or are they simply codifications of the coordinated asymmetries exhibited by the solutions of such theories? See Pooley (2003, 272–4). 9An excellent characterisation of the principle-constructive distinction is found in Balashov and Janssen (2003, 331). We have much to say about their paper in what follows. For other recent discussions of the role played by the distinction in the history and philosophy of SR, see Brown and Pooley (2001) and Brown (2003).

4 that Einstein’s comments in the 1920s on the role of the Newtonian and SR “ethers”, or space-times, indicate that he came to interpret inertial structure as a genuinely constructive element in these theories. (In our view it is unwarranted to attribute the same view to Einstein around 1905.) However, relativistic effects such as length contraction and time dilation are another matter. It is clear that in 1905, and for many subsequent years, Einstein regarded their derivation in SR as akin to the derivation in thermodynamics of, say, the existence of entropy as a thermodynamic coordinate—as being, that is to say, a necessary condition for the validity of certain phenomenological principles that themselves have only empirical robustness as their justification.10 We have discussed elsewhere Einstein’s recognition of the fact that constructive theories have more explanatory power than principle theories, as well as the misgivings that he expressed, particularly late in his life, about the appropriate- ness of his separation of kinematical and dynamical considerations in the 1905 paper (Brown and Pooley, 2001). What we wish to consider here is the question of the possibility of a fully constructive rendition of SR, and in particular the possibility of a constructive explanation of the ‘kinematical’ effects associated with length contraction and time dilation. The issues surrounding this question have been discussed recently by Bal- ashov and Janssen (2003). As will soon become clear, we take a different view to them about what might constitute a constructive version of SR. However, before addressing this issue directly, we want to return briefly to the claim that principle theories lack the explanatory power of constructive theories, for this, too, is an issue addressed by Balashov and Janssen.

10It is widely known that the fullest account given by Einstein of the claim that SR has the nature of a ‘principle-theory’ was in an article on relativity theory he was commissioned to write in 1919 for The Times of London (Einstein, 1919). Should it be thought that the popular nature of the publication and/or its date lessen the degree to which Einstein’s claim is to be taken seriously, two points might be borne in mind. First, the claim is entirely consistent with the story of Einstein’s pre-1905 struggles with the constructive approach to electrodynamics and the theory of the electron—which were based largely on the diﬃculties posed by the emergence of Planck’s constant (see below, pp. 7ﬀ). Secondly, the methodological analogy between SR and thermodynamics was mentioned by Einstein on several occasions prior to 1919. In a short paper of 1907 replying to a query of Ehrenfest on the deformable electron, he wrote: The principle of relativity, or, more exactly, the principle of relativity together with the principle of the constancy of velocity of light, is not to be conceived as a “complete system”, in fact, not as a system at all, but merely as a heuris- tic principle which, when considered by itself, contains only statements about rigid bodies, clocks, and light signals. It is only by requiring relations between otherwise seemingly unrelated laws that the theory of relativity provides addi- tional statements. . . . we are not dealing here at all with a “system” in which the individual laws are implicitly contained and from which they can be found by deduction alone, but only with a principle that (similar to the second law of the theory of heat) permits the reduction of certain laws to others. (Einstein, 1907) In a letter to Sommerfeld of 1908, Einstein wrote: The theory of relativity is not more conclusively and absolutely satisfactory than, for example, classical thermodynamics was before Boltzmann had interpreted entropy as probability. If the Michelson-Morley experiment had not put us in the worst predicament, no one would have perceived the relativity theory as a (half) salvation. Besides, I believe that we are still far from having satisfactory elementary foundations for electrical and mechanical processes.(Einstein, 1993, 50)

5 Balashov and Janssen see no problem with the idea that Einstein’s original principle-theory presentation of SR can be held to explain the phenomenon of length contraction. They write: “Understood purely as a theory of principle, SR explains this phenomenon if it can be shown that the phenomenon necessarily occurs in any world that is in accordance with the relativity postulate and the light postulate.” They concede that, in contrast to constructive-theory explanations, such a principle-theory explanation will “have nothing to say about the reality behind the phenomenon” (2003, 331). Later in their paper, which is a critical review of aspects of William Lane Craig’s recent writings in defence of presentism (Craig, 2000a,b, 2001), they take explicit issue with two claims that they attribute to Craig: (i) that SR in its 1905 form fails to provide a theory-of-principle explanation of phenomena such as length contraction and, (ii) that theory-of-principle explanations in general are deficient (2003, 332). We side with Craig on both counts, although it should be stressed that we endorse (i) for reasons quite different to those that moti- vate Craig. Before outlining our reasons for rejecting the idea that Einstein’s 1905 derivation of the Lorentz transformations can provide any sort of explanation of length contraction we mention Balashov and Janssen’s main reason for contesting (ii).11 It rests, simply, in their noting that (ii) applies equally to thermodynamics: “That in and of itself, we submit, places the relativity interpretation [i.e., Einstein’s 1905 presentation of SR] in very good company” (2003, 332). It is certainly true that Einstein’s original derivation of SR is in good company, but this company is not necessarily a company rich in explanatory resources. Bal- ashov and Janssen are prepared to admit that Einstein thought that principle theories were “inferior” to constructive theories, but this rather general claim might seem to miss the very point of Einstein’s articulation of the constructive versus principle theory distinction, and his citation of thermodynamics as a paradigm example of a principle theory. Einstein’s view (one that we share) was that principle theories were ‘inferior’ specifically in their explanatory power. His contrasting thermodynamics, as a principle theory, with statistical mechanics, as a constructive theory, was supposed to illustrate precisely that: It seems to me . . . that a physical theory can be satisfactory only when it builds up its structures from elementary foundations. (Ein- stein, 1993)12 . . . when we say we have succeeded in understanding a group of natural processes, we invariably mean that a constructive theory has been found which covers the processes in question. (Einstein, 1982, 228) It is certainly not the case that Einstein viewed principle theories as inferior in other respects. As Balashov and Janssen rightly note, their founding principles

11They also note that principle-theory derivations of particular phenomena ﬁt the covering law model of explanation, but, as they rightly concede, this “might just be another nail in the coﬃn of the covering law model” (Balashov and Janssen, 2003, 332). 12It is clear to us that in his 1908 letter, by the term “elementary foundations” Einstein means the building blocks of a constructive theory (see footnote 10 above, which contains the sentences from the letter that follow on immediately from the sentence just quoted). However, Stachel appears to think that the term refers to principles akin to those of thermodynamics (Einstein, 1989a, xxii).

6 often enjoy particularly strong empirical confirmation. Einstein is well known for having had greater confidence in the laws of thermodynamics (and, for the same reason, in SR) than in any other laws of physics. An examination of the status of length contraction in the context of Ein- stein’s 1905 treatment of SR will illustrate the way in which principle theories fail to be explanatory. Recall that in this derivation the first conclusion drawn from the two fundamental postulates is the invariance of the speed of light, that it has the same constant value in all inertial frames. This gives the ‘k-Lorentz transformations’, the Lorentz transformations up to a velocity dependent scale factor, k. What has, in effect, been shown is that if the speed of light as measured with respect to frame F 0 is to be found to be the same value as when measured with respect to the ‘resting frame’ F , then rods and clocks at rest in F 0 had better contract and dilate (with respect to frame F ) in the coordinated way that is encoded in the k-Lorentz transformations. One then appeals to the relativity principle again—the principle entails that these coordinated contrac- tions and dilations must be exactly the same function of velocity for each inertial frame—, along with the principle of spatial isotropy, in order to narrow down the deformations to just those encoded in the Lorentz transformations.13 What has been shown is that rods and clocks must behave in quite particular ways in order for the two postulates to be true together. But this hardly amounts to an explanation of such behaviour. Rather things go the other way around. It is because rods and clocks behave as they do, in a way that is consistent with the relativity principle, that light is measured to have the same speed in each inertial frame. We now return to the question of what might constitute a constructive version of SR. It is useful in this connection to start by recalling that Einstein had not adopted the principle theory route to SR by chance. He was familiar with Lorentz’s semi-constructive efforts in the 1890s to account for the null result of the 1887 Michelson-Morley experiment in terms of a postulated shape defor- mation suffered by solid bodies as a result of motion through the luminiferous ether. But by 1905 Einstein had convinced himself for a number of reasons that a systematic understanding of the non-Newtonian behaviour of moving rods and clocks based on the study of the forces holding their constituent parts together was, at that time, far too ambitious. Note that Einstein did not reject the approach initiated by Lorentz primarily because it violated the relativity principle. Although Lorentz believed in a preferred inertial frame, by 1904 the kinematics of his theory of the electron were consistent with the relativity principle. His theorem of corresponding states was based on the assumption that no experiment could be performed that would ex- hibit the presence of the ether, at least as regards effects that were up to second order in v/c; for all predictive purposes Lorentz’s theory of the electron had become compatible with the relativity principle. What instead concerned Einstein was the confused state of understanding—exacerbated by his own revolutionary

13The importance of this second application of the relativity principle (together with spatial isotropy) in Einstein’s 1905 logic was stressed in Brown (1997) and particularly in Brown and Pooley (2001). Janssen points out that in 1905 Einstein “found part of another result that had been found by Poincar´e,namely, that the Lorentz transformations form what mathematicians call a group” (Janssen, 2002a, 428). But it is important to realize that it is appeal to the relativity principle that justiﬁes the fact that the coordinate transformations form a group. The group property is essentially a postulate in Einstein’s reasoning, not a theorem.

7 hypothesis of light quanta!—of the stability of matter in terms of the dynamical forces operating at the atomic and molecular levels. By the late 1940s, a much better picture, at least in broad terms, of the cohesion of matter was available. Even so, in his 1949 Autobiographical Notes14 Einstein’s reservations about quantum mechanics apparently prevented his re- examining the constructive route to SR, despite his now articulating clear misgivings about key aspects of his 1905 principle theory approach. But what is especially striking is this. We saw above in section 1 that in 1922 Einstein referred to the SR “ether” as having an “inﬂuence on light propagation,” but in the 1949 Notes he warns against imagining that space-time intervals “are physical entities of a special type, intrinsically diﬀerent from other variables (‘reducing physics to geometry’, etc.).” Since the twenties there has been a small minority of voices—including those of Pauli, Eddington, Swann and Bell—defending, to a greater or lesser extent, the importance of a constructive, non-geometric picture of the kinematics of SR that makes no commitment to the existence of a preferred inertial frame. We have added our voices this little-known tradition (see Brown, 1993, 1997; Brown and Pooley, 2001; Brown, 2003). Recently we dubbed the approach, following a remark of John Bell (1976, 77), the “Lorentzian pedagogy”. This label has proved to have several unfortunate and highly misleading features, the worst being that any position named after Lorentz risks being misinterpreted as an endorsement of a preferred frame.15 A more appropriate label would be the dynamical interpretation.16 But as one of us has noted (Brown, 1997) the ‘space-time theory’ approach developed principally by philosophers in North America in recent decades— the view of SR that is encapsulated in Friedman’s 1983 book Foundations of Space-Time Theories—also could be interpreted as a constructive theory in Einstein’s sense, where it is precisely the Minkowski geometry that provides the explanatory deep structure. An explicit defence of this position has recently been given by Balashov and Janssen, to which we now turn.

4 The explanation of length contraction

How are we to explain length contraction in SR? One needs to be careful about what, exactly, is taken to stand in need of an explanation. Balashov and Janssen’s initial characterization of the constructive-theory explanation of the space-time interpretation runs as follows: length contraction is explained by showing that two observers who are in relative motion to one another and therefore use different sets 14See Einstein (1969). 15The remaining unfortunate features are these. First, the pedagogic dimension offered by Bell’s simple atomic model displaying motion-induced relativistic contraction is not the whole story, as Bell himself recognized (see also Brown and Pooley, 2001). Secondly, as has been argued recently in Brown (2003), the term “FitzGeraldian pedagogy” would be historically more appropriate. Finally, in so far as there is a connection with Lorentz’s thinking, it is only his post-1905 formulation of the electron theory, in which Lorentz had learnt from Einstein how correctly to interpret the Lorentz transformations (see Janssen, 2002a, 8) that is relevant—but shorn of the privileged frame! 16Such an approach does not appear (under any label) within the recent taxonomy of interpretations of SR produced by Craig, and endorsed, with qualification, by Balashov and Janssen.

8 of space-time axes disagree about which cross-sections of the ‘world- tube’ of a physical system give the length of the system. (2003, 331) Here we are asked to contemplate a single rod. What is to be explained is how it is possible that this single rod comes to be assigned two different lengths when measured with respect to two inertial frames. Note that the relativity of simultaneity—that two different cross-sections of the rod are involved—plays a crucial role.17 In his contribution to this volume (Saunders, 2003, xxx), Saunders considers two rods, R and S, in relative inertial motion. Specific features of Minkowski geometry are appealed to in an explanation of why, relative to surfaces of simultaneity orthogonal to the world-tube of R, S is shorter than R whereas, relative to surfaces of simultaneity orthogonal to the world-tube of S, it is R that is shorter than S.18 In our opinion these constitute perfectly acceptable explanations (perhaps the only acceptable explanations) of the explananda in question. But it is far from clear that they qualify as constructive explanations.19 What is being assumed in both cases is that the rod(s) being measured, and the rods and clocks doing the measuring, all satisfy the constraints of Minkowskian geometry. The explanations point out that if objects obeying these constraints have certain geometrical features, then it follows, as a simple consequence of the mathematics of Minkowskian geometry, that they will have certain other features. The geometrical features of the objects that are assumed, and appealed to, in these explanations are similar in status to the postulates of principle theories. They do not, directly, concern the details of the bodies’ microphysical constitution. Rather they are about aspects of their (fairly) directly observable macroscopic behaviour. And this reflection prompts an obvious question: why do these objects obey the constraints of Minkowski geometry?20 It is precisely this question that calls out for a constructive explanation. What sort of an answer might be given? The following quote from Friedman helps to delineate the options. In dis- cussing Poincaré’spreference for “the Lorentz-Fitzgerald version of an ‘aether’ theory” over Einstein’s formulation of SR he writes: . . . the crucial difference between the two theories, of course, is that the Lorentz contraction, in the former theory, is viewed as a result of the (electromagnetic) forces responsible for the microstructure of

17In a recent manuscript, Petkov claims to show that “no forces are involved in the explanation of the Lorentz contraction” (Petkov, 2002, 6). His argument involves consideration of essentially the same scenario considered by Balashov and Janssen. And, of course, those who believe (like us) that in some explanatory contexts it is correct to invoke forces would not do so when comparing one cross-section of the world tube of a rod with another cross-section of the same rod. Rather, forces are relevant, for example, when comparing, relative to a ﬁxed inertial frame and standard of simultaneity, two otherwise identical rods that are in diﬀerent states of motion relative to this frame of reference. 18An analogous scenario is also considered by Janssen (2002b, 499–500). 19It should be stressed that Saunders does not claim that the explanation he sketches is a constructive-theory explanation. 20Note that this question arises for someone with no prior expectations about how bodies in motion ‘should’ behave; pace Balashov and Janssen (2003, 340), the question need not be understood as asking “why do these objects obey the constraints of Minkowski geometry rather than those of Newtonian space-time?”

9 matter in the context of Lorentz’s theory of the electron, whereas this same contraction, in Einstein’s theory, is viewed as a direct reflection—independent of all hypotheses concerning microstructure and its dynamics—of a new kinematical structure for space and time involving essential relativized notions of duration, length, and simultaneity. In terms of Poincaré’shierarchical conception of the sciences, then, Poincarélocates the Lorentz contraction (and the Lorentz group more generally) at the level of experimental physics, while keeping Newtonian structure at the next higher level (what Poincarécalls mechanics) completely intact. Einstein, by contrast, locates the Lorentz contraction (and the Lorentz group more generally) at precisely this next higher level, while postponing to the future all further discussion of the physical forces and material structures actually responsible for the physical phenomenon of rigidity. The Lorentz contraction, in Einstein’s hands, now receives a direct kinematical interpretation. (Friedman, 2002, 211–2) The talk of a preference for one theory over the the other might suggest that we are dealing with two incompatible, rival viewpoints. On one side one has a truly constructive space-time interpretation of SR, involving the postulation of the structure of Minkowski space-time as an ontologically autonomous element in the models of the phenomena in question. In this picture, length contraction is to be given a constructive explanation in terms of Minkowski space-time because complex material bodies are constrained (somehow!) to “directly reflect” its structure, in a way that is “independent of all hypotheses concerning microstructure and its dynamics.”21 If one were to adopt such a viewpoint there would seem to be little room left for the alternative viewpoint, according to which the explanation of length contraction is ultimately to be sought in terms of the dynamics of the microstructure of the contracting rod. In fact, it is not clear that Friedman has these two opposing pictures in mind. Although he claims that Poincaré keeps Newtonian structure at the level of ‘mechanics’, if one is committed to the idea that Lorentz contraction is the result of the forces responsible for the microstructure of matter then one should, in our opinion, believe that Minkowskian, rather than Newtonian, structure is the appropriate kinematics for mechanics. In our view, the appropriate structure is Minkowski geometry precisely because the laws of physics, including those to be appealed to in the dynamical explanation of length contraction, are Lorentz covariant. Equally one can postpone (as Einstein did) the detailed investigation into the forces and structures actually responsible for the phenomena that are paradigmatic of space-time’s Minkowskian geometry without thereby relinquish- ing the idea that these forces and structures are, indeed, “actually responsible” for the phenomena in question and, hence, (we go further in suggesting) for space-time having the structure that it has. Saunders is critical of the Lorentzian pedagogy because he takes it to require that the investigation of dynamical phenomena is to be referred to a single (though arbitrary) frame of reference. It is true that Bell was concerned to extol the virtues of working wholly within a single frame. But on this score his

21The thesis that Minkowski spacetime cannot act in this way as an explanans in a constructive version of SR was put forward in Brown (1997) and further defended in Brown and Pooley (2001) and Brown (2003).

10 point was primarily a pedagogic one. His point was not that one is required to work in a single framed, but that one always can work in a single frame. In particular, he was concerned to show that, if one knew the laws of physics with respect to a given frame one could, at least in principle, derive how they should be described with respect to other frames.22 Bell believed that exploiting the perspective one gains from working with respect to a single frame best allows one to discern the great continuity that exists between relativity and the physics that predated Einstein’s 1905 paper. As such, the single-frame perspective is a useful antidote to misapprehensions about relativity that arise when one focusses solely on the discontinuities. But focus on describing all phenomena with respect to a single frame is just one part of Bell’s message. Moreover, it is not that part which forms the essential element in the position we have called the dynamical interpretation. What is definitive of this position is the idea that constructive explanation of ‘kinematic’ phenomena involves investigation of the details of the dynamics of the complex bodies that exemplify the kinematics. And it seems that Saunders agrees on this score. Given a word-line that represents the possible trajectory of the end point of a small rod, and given a single point that is meant to represent the other end of the rod at some particular moment, there is, from the point of view of Minkowski geometry, a particularly natural construction of a second curve through the single point. The two curves together define the possible world-tube of an extended body, a world-tube that, from the point of view of Minkowski geometry, is particularly natural. According to Saunders “it is this construction that needs a dynamical underpinning: why do stable bodies, sufficiently small in size, have world-tubes with this geometry?” (Saunders, 2003, xxx). This, we claim, is precisely the type of question that the dynamical interpretation of SR seeks to address. Lit- tle hangs on whether the dynamical underpinning is spelled out with respect to a particular frame, or whether the solution is given in some sophisticated, coordinate independent way. What is important is that particular laws—a specific quantum field theory—could be solved and the solutions shown to have the requisite geometrical properties.23

22If the laws known with respect to a given frame are in fact (though not known to be) Lorentz covariant, one will derive that the rods and clocks at rest in another frame will be contracted and dilated relative to one’s own: one will derive that a Lorentz transformation is the correct coordinate transformation relating the two frames. One can then go on to investigate how phenomena in general are to be described relative to this frame, and to derive that these descriptions will obey laws of exactly the same form as do descriptions with respect to one’s own frame. One will have thereby derived the Lorentz covariance of the laws (see Bell, 1987, 75–6; cf. Swann, 1941). 23It is perhaps worth mentioning here one common objection to any approach that seeks to reduce non-dynamical space-time structure, such as that of SR, to the symmetries of the laws governing matter. According to the objection such an approach is constrained to use special coordinates (in which the laws take their canonical form) because otherwise the geometric structure, in the form of the Minkowski metric and its connection coeﬃcients, appears explicitly in the laws. Two things should be said in response to this objection. First, the objection surely is not that defenders of the reductive account are okay so long as they place restrictions on admissible coordinate systems. If the reductive account is tenable at all, then it can countenance the use of arbitrary coordinate systems. (Perhaps it could be argued that the supporter of the reductive account faces an obligation to provide an alternative formulation of the laws as written with respect to arbitrary coordinates, in which the secondary status of the geometrical structure is clear.) Secondly, even if one is inclined to take the appearance of the Minkowski metric in the

11 We have been arguing that the truly constructive explanation of length contraction involves solving the dynamics governing the structure of the complex material body that undergoes contraction. There are, of course, many contexts in which such an explanation may not be appropriate, contexts that call for a purely geometrical explanation. What we wish to stress is (i) that such geometrical explanations are not constructive theory explanations in Einstein’s sense and (ii) that there are contexts, and questions, to which the dynamical story is appropriate. There is one final, important, area of disagreement between ourselves and Balashov and Janssen to map out. But before we do so, it will be instructive to acknowledge that in many contexts, perhaps in most contexts, one should not appeal to the details of the dynamics governing the microstructure of bodies exemplifying relativistic effects when one is giving a constructive explanation of them.24 Granted that there are stable bodies, it is sufficient for these bodies to undergo Lorentz contraction that the laws (whatever they are) that govern the behaviour of their microphysical constituents are Lorentz covariant. It is the fact that the laws are Lorentz covariant, one might say, that explains why the bodies Lorentz contract. To appeal to any further details of the laws that govern the cohesion of these bodies would be a mistake. Elsewhere we have dubbed this view the “truncated” Lorentzian pedagogy (Brown and Pooley, 2001, 261). It is worth making two points about it. First, to explain why there are any bodies at all that conform to Minkowskian geometry one needs to appeal to more than Lorentz covariance. One needs to demonstrate the possibility of stable material configurations, and the constructive explanation of this will involve a more complete dynamical analysis. Secondly, one might be tempted to deny that explanations which appeal to an explanans as non-concrete as the symmetries of the laws are genuinely constructive explanations. In other words, it turns out that there are even fewer contexts than one might have at first supposed in which length contraction stands in need of a constructive-theory explanation.

5 Minkowski space-time: the cart or the horse?

But if it is often sufficient to appeal to Lorentz covariance to give a dynamical explanation of length contraction, is that where explanations should stop? It is here that Balashov and Janssen see a further, constructive role for the geometry of space-time. They ask: ... does the Minkowskian nature of space-time explain why the forces holding a rod together are Lorentz invariant or the other way around? Our intuition is that the geometrical structure of space(- time) is the explanans here and the invariance of the forces the explanandum. To switch things around, our intuition tells us, is laws when written generally covariantly as a reason to afford it a primitive ontological status, one is still obliged to tell some story about how and why material systems reflect its structure in their macroscopic behaviour. What could this story be, other than the dynamical one? (In a recent discussion, Butterfield also recognises the existence of such an obligation, which, in his terminology, is an obligation to answer to the “consistency problem” (Butterfield, 2001, §2.1.2).) 24We thank Michel Janssen for reminding us of this point.

12 putting the cart before the horse. (Balashov and Janssen, 2003, 340–1) The same issue was raised some years ago in Brown (1993) and, particularly, Brown (1997), but there the opposite view to Balashov and Janssen’s was taken as to what was to be regarded as the cart and what the horse. It is worth recalling that Balashov and Janssen’s target is the particular neo- Lorentzian interpretation of SR advocated by Craig. This is an interpretation in which space-time structure is supposed to be Newtonian and in which there is supposed to be a preferred frame, consistent with Craig’s commitment to a tensed theory of time. Balashov and Janssen’s claim is that the space-time interpretation has a definite explanatory advantage over this neo-Lorentzian interpretation when it comes to the Lorentz covariance of the laws governing the behaviour of matter: In the former, Lorentz invariance reflects the structure of the space-time posited by the theory. In the latter, Lorentz invariance is a property accidentally shared by all laws effectively governing systems in Newtonian space and time. . . In the neo-Lorentzian interpretation it is, in the final analysis, an unexplained coincidence that the laws effectively governing different sorts of matter all share the property of Lorentz invariance, which originally appeared to be nothing but a peculiarity of the laws governing electromagnetic fields. In the space-time interpretation this coincidence is explained by tracing the Lorentz covariance of all these different laws to a common origin: the space-time structure posited in this interpretation (Janssen [1995], [2002]). . . No matter how the argument is made, the point is that there are brute facts in the neo-Lorentzian interpretation that are explained in the space-time interpretation. As Craig (p. 101) writes (in a different context): ‘if what is simply a brute fact in one theory can be given an explanation in another theory, then we have an increase in intelligibility that counts in favor of the second theory.’ We agree that in Craig’s neo-Lorentzian interpretation of SR, and according to our preferred dynamical interpretation, the Lorentz covariance of all the fundamental laws of physics is an unexplained brute fact. This, in and of itself, does not count against the interpretations: all explanation must stop some- where. What is required if the so-called space-time interpretation is to win out over the dynamical interpretation (and Craig’s neo-Lorentzian interpretation) is that it offers a genuine explanation of Lorentz covariance. This is what we dis- pute. Talk of Lorentz covariance “reflecting the structure of space-time posited by the theory” and of “tracing the invariance to a common origin” needs to be fleshed out if we are to be given a genuine explanation here—something akin to the explanation of inertia in general relativity (see section 1 above). Otherwise we simply have yet another analogue of Moliere’s dormative virtue. In fact Balashov and Janssen’s own example can be turned against them. Craig’s neo-Lorentzian interpretation is precisely an example of theory in which the symmetries of spacetime structure are not reflected in the symmetries of the laws governing matter. Balashov and Janssen do not question the coherence of this theory (as we would). Rather they seek to rule it out on the grounds of its

13 explanatory deficiencies when compared to their preferred theory. This shows that, as matter of logic alone, if one postulates spacetime structure as a self- standing, autonomous element in one’s theory, it need have no constraining role on the form of the laws governing the rest of content of the theory’s models.25 So how is its influence on these laws supposed to work? Unless this question is answered, space-time’s Minkowskian structure cannot be taken to explain the Lorentz covariance of the dynamical laws. From our perspective, of course, the direction of explanation goes the other way around. It is the Lorentz covariance of the laws that underwrites the fact that the geometry of space-time is Minkowskian. It is for this reason that we can rule out the sort of mismatch between space-time symmetries and dynamical symmetries that are a feature of Craig’s interpretation, and that so trouble Balashov and Janssen. Balashov and Janssen acknowledge that some of their readers will have this ‘relationist’ intuition. Remarkably they claim that this does not weaken their point! In a footnote, they admit that, for the relationalist, the Lorentz covariance of the laws “in a sense does seem to explain” why space-time structure in Minkowskian. (We, of course, see no reason for their qualifications here.) But, they go on to assert that the relationalist should nonetheless view, for example, the Euclidean nature of space as explaining why the forces holding Cyrano’s nose together are invariant under rotations rather than vice versa (Balashov and Janssen, 2003, 341, fn 11; cf. 340). As far as we can see, this amounts to bald assertion. We happily concede that there are many contexts in which the Euclidean nature of space is the appropriate explanation of the behaviour of Cyrano’s nose. But we insist that there are others in which it is appropriate to appeal to the Euclidean symmetries of the forces at work to explain the same behaviour. And we simply deny that the Euclidean nature of space can ever be cited as a genuine explanation of these symmetries; this would be to put the cart before the horse. A more sustained discussion of Minkowski space-time’s providing a putative common origin for the “unexplained coincidence” in Lorentz’s theory that both matter and fields are governed by Lorentz covariant laws, is to be found in Janssen’s detailed recent analysis of the differences between the Einstein and Lorentz programs (Janssen, 2002a). It is also covered in his wider investigation of ‘common origin inferences’ in the history of science (Janssen, 2002b, 497– 507). In our view, neither of these papers succeed in clarifying how space-time structure can act as a “common origin” of otherwise unexplained coincidences. One might, for example, go so far as to agree that all particular instances of paradigmatically relativistic kinematic behaviour are traceable to a common origin: the Lorentz covariance of the laws of physics. But Janssen wants us to go further. He wants us to then ask after the common origin of this universal Lorentz covariance. It is his claim that this can be traced to the space-time structure posited by Minkowski that is never clarified. For example, immediately after making this claim in Janssen (2002b), he writes: 25See in this connection Brown (1993). It might be useful to recall here the example of the approach to Einstein’s field equations in GR based on the the introduction of a spin-2 field on flat Minkowski space-time (for references, see Preskill and Thorne, 1999, xiii–iv). The operational significance of the background space-time in this theory is not the same as that in SR.

14 In Minkowski space-time, the spatio-temporal coordinates of different observers are related by Lorentz transformations rather than Galilean transformations. Any laws for systems in Minkowski space- time must accordingly be Lorentz invariant. There is an dangerous ambiguity lurking here. The state of affairs described in the first sentence cannot be held to explain the Lorentz covariance of the laws (surely the claim that Janssen intends). But one can take the state of affairs described in the first sentence as evidence for the Lorentz covariance of the laws.26 The passage quoted is true only if one understands it as making such an evidentiary claim. And as such, it is (essentially) an unexceptionable statement of Einstein’s 1905 reasoning.27 We hope to have made it clear why we do not believe that Minkowski space- time can play the constructive explanatory role that Balashov and Janssen would have it serve. What needs to be stressed is that this conclusion is appropriate not only for those who adopt an eliminative relationalist stance towards the ontology of space-time, and not only in the context of theories with fixed, absolute space-time structure. As we argued in Brown and Pooley (2001), even when one’s ontology includes substantival space-time structure, the symmetries of the laws governing material systems are still crucial in such structure gaining operational chronogeometric significance. As we wrote elsewhere: Despite the fact that in GR one is led to attribute an independent real existence to the metric field, the general relativistic explanation of length contraction and time dilation is simply the dynamical one we have urged in the context of special relativity. (Brown and Pooley, 2001, 271)

Acknowledgements

We are grateful to Michel Janssen and Simon Saunders for discussion. This paper was composed during Oliver Pooley’s tenure of a British Academy Postdoc- toral Fellowship; he gratefully acknowledges the support of the British Academy.

26Cf. Janssen’s own distinction between ‘explanatory’ and ‘evidentiary’ uses of “because” (Janssen, 2002b, 459). 27It is also worth noting that a curious picture of Einstein’s pre-Minkowskian work emerges in Janssen (2002a). Janssen stresses that “In Einstein’s special theory of relativity the Lorentz invariance of these different laws [of matter and fields] is traced to a common origin” (p. 6) and that “Einstein recognized that Lorentz invariance reflects a new space-time structure” (p. 9). But Janssen himself acknowledges that in 1905 Einstein never talks about space-time, and that his initial reaction to Minkowski’s geometrization of his 1905 theory was negative (p. 9). The careful reader of Janssen’s study would be forgiven for thinking that Einstein misunderstood his own theory in 1905, or at least its real point of departure from Lorentz’s program. Our position, on the other hand, is that is that Einstein knew pretty well what he was doing in 1905. In providing a principle theory approach to deriving the Lorentz transformations, and hence the non-classical behaviour of rods and clocks, he was re-systematizing, and giving a different emphasis to, aspects of the work of Lorentz and Poincaré,but not providing a revolutionary new stance. For Einstein himself, the real revolution in his 1905 annus mirabilis was his light quantum hypothesis, as is well known. It has been aptly noted by Staley (1998, 272–4) that despite the fact that physicists seldom distinguished between Lorentz’s and Einstein’s formulation of the electron theory in the years immediately following 1905, Einstein did not seek to redress this situation—indeed he even referred to “the theory of Lorentz and Einstein” in 1906 (though admittedly in somewhat special circumstances).

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